Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(17,18)(19,20)(21,24)(22,23);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,18,19,24,22,23,21,20);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(18,23)(20,24);(2,6)(5,7)(10,14)(13,16)(17,19,22,21)(18,24,23,20);(2,6)(5,7)(17,20,19,18,22,24,21,23)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72313739, #72313776, #72313813, … |
| maximal normal subgroup of | #72313739, #72313776, #72313813, … |
| maximal quotient of | #72320164, #72320210 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507399);